which belongs to ??\mathfrak{S}, may be taken for the upper bound of CCC, since it clearly is an ideal of RRR and disjoint with SSS. Because ??\mathfrak{S} thus is inductively ordered with respect to “⊆\subseteq”, our assertion follows from Zorn’s lemma.

Definition. The maximal elements in the Lemma are prime ideals of the commutative ring.